Quantum entanglement

Quantum entanglement is a physical phenomenon that occurs when pairs or groups of particles are generated or interact in ways such that the quantum state of each particle cannot be described independently—instead, a quantum state may be given for the system as a whole.

When the quantum system contains more than one particle, the superposition principle gives rise to the phenomenon of entanglement. It is now not just a particle interfering with itself—it is a system interfering with itself: an entangled system. Amazingly enough, Erwin Schrödinger himself realized that particles or photons produced in a process that links them together will be entangled, and he actually coined the term entanglement, both in his native German and in English. Schrödinger discovered the possibility of entanglement in 1926, when he did his pioneering work on the new quantum mechanics, but he first used the term entanglement in 1935, in his discussion of the Einstein, Podolsky, and Rosen (EPR) paper.

Roger Ebert, in Life Itself : A Memoir (2011), Ch. 54 : How I Believe In God

When two systems, of which we know the states by their respective representation, enter into a temporary physical interaction due to known forces between them and when after a time of mutual influence the systems separate again, then they can no longer be described as before, viz., by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics.

The phenomenon of entanglement is the essentialfact of quantum mechanics, the fact that makes it so different from classical physics. It brings into question our entire understanding about what is real in the physicalworld. Our ordinary intuition about physical systems is that if we know everything about a system, that is, everything that can in principle be known, then we know everything about its parts. If we have complete knowledge of the condition of an automobile, then we know everything about its wheels, its engine, its transmission, right down to the screws that hold the upholstery in place. It would not make sense for a mechanic to say, “I know everything about your car but unfortunately I can’t tell you anything about any of its parts.”
But that’s exactly what Einstein explained to Bohr — in quantum mechanics, one can know everything about a system and nothing about its individual parts — but Bohr failed to appreciate this fact. I might add that generations of quantum textbooks blithely ignored it.

Leonard Susskind, in the Preface of Quantum Mechanics: The Theoretical Minimum (2014) by Leonard Susskind and Art Friedman

There is a troubling weirdness about quantum mechanics. Perhaps its weirdest feature is entanglement, the need to describe even systems that extend over macroscopic distances in ways that are inconsistent with classical ideas.